L(s) = 1 | + (0.540 + 0.841i)2-s + (0.989 + 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (−0.755 + 0.654i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.540 + 0.841i)12-s + (0.755 + 0.654i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + (0.281 + 0.959i)18-s + (0.415 − 0.909i)19-s + ⋯ |
L(s) = 1 | + (0.540 + 0.841i)2-s + (0.989 + 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (−0.755 + 0.654i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (−0.540 + 0.841i)12-s + (0.755 + 0.654i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (0.909 − 0.415i)17-s + (0.281 + 0.959i)18-s + (0.415 − 0.909i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.080714052 + 1.203932226i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.080714052 + 1.203932226i\) |
\(L(1)\) |
\(\approx\) |
\(1.292090645 + 0.8710866815i\) |
\(L(1)\) |
\(\approx\) |
\(1.292090645 + 0.8710866815i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 3 | \( 1 + (0.989 + 0.142i)T \) |
| 7 | \( 1 + (-0.755 + 0.654i)T \) |
| 11 | \( 1 + (-0.841 - 0.540i)T \) |
| 13 | \( 1 + (0.755 + 0.654i)T \) |
| 17 | \( 1 + (0.909 - 0.415i)T \) |
| 19 | \( 1 + (0.415 - 0.909i)T \) |
| 29 | \( 1 + (-0.415 - 0.909i)T \) |
| 31 | \( 1 + (-0.142 - 0.989i)T \) |
| 37 | \( 1 + (-0.281 + 0.959i)T \) |
| 41 | \( 1 + (-0.959 + 0.281i)T \) |
| 43 | \( 1 + (-0.989 - 0.142i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.755 - 0.654i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.540 - 0.841i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.909 - 0.415i)T \) |
| 79 | \( 1 + (-0.654 + 0.755i)T \) |
| 83 | \( 1 + (0.281 - 0.959i)T \) |
| 89 | \( 1 + (-0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.281 + 0.959i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.31970240165594619884285690319, −28.16614495518225499822124839457, −27.01248050715662813079288902400, −25.960670741205725245719198681241, −25.027281782315803148926601604729, −23.56252348112538679251992599845, −23.0142785265851386434938630851, −21.58428857902324058045819491510, −20.56320054337773063915075217450, −20.04498964488253435354042320291, −18.91337794457542750445889721115, −18.1085427041094559942829141071, −16.16122967394435830842089677739, −15.05037806938861799737974062852, −13.996453520781944297740187344632, −13.07979161826336629899816377912, −12.3792759928082878873842606280, −10.47880812134934656395139925920, −9.94929626808068694852672356315, −8.51670982661109412957832288490, −7.1287968388849501330942616627, −5.49583561726702936724440563976, −3.819064797792719766422887719252, −3.08874423632654308614938348207, −1.49138281670151486374539702654,
2.683020104500789345828229770363, 3.64154659790839762031299623903, 5.17469910497437868199765869492, 6.48738529115959306888707366897, 7.779393368580819740660666617258, 8.77709714311247151007089944717, 9.75995155480842915429678266988, 11.72910314192758358895762694228, 13.18424444385115859907524696549, 13.64254505761390653925220222155, 14.98741025831790030238174228979, 15.78269582858264990085333555064, 16.550882515495520978859108166071, 18.357467414650447358208419264678, 19.00050708785757794351438590097, 20.621945840226299238185495616701, 21.417012094855110993599840183517, 22.38015153411735323780440880978, 23.66672242505453264934387454566, 24.54389846043861478072089867580, 25.69016104084876182664939937044, 26.035805675236686922145044634402, 27.11692643904601036730132131776, 28.49522322839824027655339226173, 29.9117743706099284155122112981